Spectral representation of absolutely minimum attaining unbounded normal operators

Abstract

Let $T:D(T)\to H_2$ be a densely defined closed operator with domain $D(T)\subset H_1$. We say $T$ to be absolutely minimum attaining if for every closed subspace $M$ of $H_1$, the restriction operator $T{|}_M:D(T)\cap M\to H_2$ attains its minimum modulus $m(T{|}_M)$. That is, there exists $x\in D(T)\cap M$ with $\Vert x\Vert =1$ and $\Vert T(x)\Vert =\inf \{\Vert T(m)\Vert :m\in D(T)\cap M:\Vert m\Vert =1\}$. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent.

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